How about a one-liner in JS ES10 / ES2019 ?

Making use of Object.entries() and Object.fromEntries():

let newObj = Object.fromEntries(Object.entries(obj).map(([k, v]) => [k, v * v]));

The same thing written as a function:

function objMap(obj, func) {
  return Object.fromEntries(Object.entries(obj).map(([k, v]) => [k, func(v)]));
}

// To square each value you can call it like this:
let mappedObj = objMap(obj, (x) => x * x);

This function uses recursion to square nested objects as well:

function objMap(obj, func) {
  return Object.fromEntries(
    Object.entries(obj).map(([k, v]) => 
      [k, v === Object(v) ? objMap(v, func) : func(v)]
    )
  );
}

// To square each value you can call it like this:
let mappedObj = objMap(obj, (x) => x * x);

With ES7 / ES2016 you can't use Objects.fromEntries, but you can achieve the same using Object.assign in combination with spread operators and computed key names syntax:

let newObj = Object.assign({}, ...Object.entries(obj).map(([k, v]) => ({[k]: v * v})));

ES6 / ES2015 Doesn't allow Object.entries, but you could use Object.keys instead:

let newObj = Object.assign({}, ...Object.keys(obj).map(k => ({[k]: obj[k] * obj[k]})));

ES6 also introduced for...of loops, which allow a more imperative style:

let newObj = {}

for (let [k, v] of Object.entries(obj)) {
  newObj[k] = v * v;
}

array.reduce()

Instead of Object.fromEntries and Object.assign you can also use reduce for this:

let newObj = Object.entries(obj).reduce((p, [k, v]) => ({ ...p, [k]: v * v }), {});

Inherited properties and the prototype chain:

In some rare situation you may need to map a class-like object which holds properties of an inherited object on its prototype-chain. In such cases Object.keys() and Object.entries() won't work, because these functions do not include the prototype chain.

If you need to map inherited properties, you can use for (key in myObj) {...}.

Here is an example of such situation:

const obj1 = { 'a': 1, 'b': 2, 'c': 3}
const obj2 = Object.create(obj1);  // One of multiple ways to inherit an object in JS.

// Here you see how the properties of obj1 sit on the 'prototype' of obj2
console.log(obj2)  // Prints: obj2.__proto__ = { 'a': 1, 'b': 2, 'c': 3}

console.log(Object.keys(obj2));  // Prints: an empty Array.
console.log(Object.entries(obj2));  // Prints: an empty Array.

for (let key in obj2) {
  console.log(key);              // Prints: 'a', 'b', 'c'
}

However, please do me a favor and avoid inheritance. :-)